Calabi-Yau pointed Hopf algebras of finite Cartan type Yinhuo Zhang - - PowerPoint PPT Presentation

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Calabi-Yau pointed Hopf algebras

• f finite Cartan type

Yinhuo Zhang University of Hasselt Noncommutative Algebraic Geometry 2011 Shanghai Workshop

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Outline

Definition of Calabi-Yau algebras Pointed Hopf algebras The Calabi-Yau property of U(D, λ) The Calabi-Yau property of Nichols algebras of finite Cartan type joint work with Xiaolan Yu

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Notations

❦ is a fixed algebraically closes field with characteristic 0. ❦

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Notations

❦ is a fixed algebraically closes field with characteristic 0. All vector spaces and algebras are assumed to be over ❦.

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Notations

❦ is a fixed algebraically closes field with characteristic 0. All vector spaces and algebras are assumed to be over ❦. All Hopf algebras mentioned are assumed to be Hopf algebras with bijective antipodes.

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Notations

❦ is a fixed algebraically closes field with characteristic 0. All vector spaces and algebras are assumed to be over ❦. All Hopf algebras mentioned are assumed to be Hopf algebras with bijective antipodes. Given an algebra A, we write Aop for the opposite algebra of A and Ae for the enveloping algebra A ⊗ Aop of A.

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Notations

Let A be an algebra and M an A-A-bimodule. For algebra automorphisms σ and τ, the bimodule σMτ is defined by a · m · b := σ(a)mτ(b).

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Notations

Let A be an algebra and M an A-A-bimodule. For algebra automorphisms σ and τ, the bimodule σMτ is defined by a · m · b := σ(a)mτ(b).

When σ or τ is the identity map, we shall simply omit it. For example, 1Mτ is denoted by Mτ.

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Notations

Let A be an algebra and M an A-A-bimodule. For algebra automorphisms σ and τ, the bimodule σMτ is defined by a · m · b := σ(a)mτ(b).

When σ or τ is the identity map, we shall simply omit it. For example, 1Mτ is denoted by Mτ. We have Aτ ∼ = τ −1A as A-A-bimodules.

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Definition of Calabi-Yau algebras

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Calabi-Yau algebras

(Ginzburg) An algebra A is called a Calabi-Yau algebra of dimension d if

(i) A is homologically smooth. That is, A has a bounded resolution of finitely generated projective A-A-bimodules. (ii) There are A-A-bimodule isomorphisms Exti

Ae(A, Ae) ∼

=

• 0,

i = d; A, i = d.

In the following, Calabi-Yau will be abbreviated to CY for short.

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Examples of Calabi-Yau algebras

The polynomial algebra ❦[x1, · · · , xn] with n variables is a CY algebra of dimension n. (Berger) Any Sridharan enveloping algebra of an n-dimensional abelian Lie algebra is a CY algebra of dimension n. (Bocklandt) Let A be the algebra ❦x0, x1, x2, x3/I, where the ideal I is generated by the following relations x0x1 − x1x0 − α(x2x3 + x3x2), x0x1 + x1x0 − (x2x3 − x3x2), x0x2 − x2x0 − β(x3x1 + x1x3), x0x2 + x2x0 − (x3x1 − x1x3), x0x3 − x3x0 − γ(x1x2 + x2x1), x0x3 + x3x0 − (x1x2 − x2x1),

α + β + γ + αβγ = 0 and (α, β, γ) / ∈ {(α, −1, 1), (1, β, −1), (−1, 1, γ)}.

The algebra A is a 4-dimensional Sklyanian algebra and a CY algebra of dimension 4.

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Dualizing complexes

(Yekutieli) Let A be a Noetherian algebra. Roughly speaking, a complex R ∈ Db(Ae) is called a dualizing complex if the functor RHomA(−, R) : Db

fg(A) → Db fg(Aop)

is a duality, with adjoint RHomAop(−, R). Here Db

fg(A) is the full triangulated subcategory of the derive

category D(A) of A consisting of bounded complexes with finitely generated cohomology modules.

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Dualizing complexes

(Yekutieli) Let A be a Noetherian algebra. Roughly speaking, a complex R ∈ Db(Ae) is called a dualizing complex if the functor RHomA(−, R) : Db

fg(A) → Db fg(Aop)

is a duality, with adjoint RHomAop(−, R). Here Db

fg(A) is the full triangulated subcategory of the derive

category D(A) of A consisting of bounded complexes with finitely generated cohomology modules. Example: The complex R = A is a dualizing complex over A if and only if A is a Gorenstein ring (i.e. A has finite injective dimension as left and right module over itself).

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Rigid Dualizing complexes

Dualizing complexes are not unique.

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Rigid Dualizing complexes

Dualizing complexes are not unique. (Van den Bergh) Let A be a Noetherian algebra. A dualizing complex R over A is called rigid if RHomAe(A, AR ⊗ RA) ∼ = R in D(Ae).

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Rigid Dualizing complexes

Theorem 1 (Van den Bergh, Brown-Zhang) Let A be a Noetherian algebra. Then the following two conditions are equivalent: (1) A has a rigid dualizing complex R = Aψ[s], where ψ is an algebra automorphism and s ∈ Z. (2) A has finite injective dimension d and there is an algebra automorphism φ such that Exti

Ae(A, Ae) ∼

=

• 0,

i = d; Aφ i = d as A-A-bimodules. In this case, φ = ψ−1 and s = d.

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Rigid Dualizing complexes

Theorem 1 (Van den Bergh, Brown-Zhang) Let A be a Noetherian algebra. Then the following two conditions are equivalent: (1) A has a rigid dualizing complex R = Aψ[s], where ψ is an algebra automorphism and s ∈ Z. (2) A has finite injective dimension d and there is an algebra automorphism φ such that Exti

Ae(A, Ae) ∼

=

• 0,

i = d; Aφ i = d as A-A-bimodules. In this case, φ = ψ−1 and s = d. Corollary 2 A Noetherian algebra A is CY of dimension d if and only if A is homologically smooth and has a rigid dualizing complex A[d].

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Artin-Schelter Gorenstein algebras

Let A be a Noetherian augmented algebra with a fixed augmentation map ε : A → ❦. The algebra A is said to be AS-Gorenstein, if

(i) injdim AA = d < ∞, where injdim stands for injective dimension. (ii) dim Exti

A(A❦, AA) =

• 0,

i = d; 1, i = d. (iii) the right version of the conditions (i) and (ii) hold.

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Artin-Schelter Gorenstein algebras

Let A be a Noetherian augmented algebra with a fixed augmentation map ε : A → ❦. The algebra A is said to be AS-Gorenstein, if

(i) injdim AA = d < ∞, where injdim stands for injective dimension. (ii) dim Exti

A(A❦, AA) =

• 0,

i = d; 1, i = d. (iii) the right version of the conditions (i) and (ii) hold.

An AS-Gorenstein algebra A is said to be regular if in addition, the global dimension of A is finite.

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Examples

A finite dimensional Hopf algebra H is AS-Gorenstein. Since a finite dimensional Hopf algebra H is Frobenius, injdimHH =injdimHH = 0, dim HomH(H❦, HH) = dim HomH(❦H, HH) = 1.

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Examples

A finite dimensional Hopf algebra H is AS-Gorenstein. Since a finite dimensional Hopf algebra H is Frobenius, injdimHH =injdimHH = 0, dim HomH(H❦, HH) = dim HomH(❦H, HH) = 1. Let g be a Lie algebra of dimension d. Then the universal enveloping algebra U(g) is AS-regular and of global dimension d.

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Examples

A finite dimensional Hopf algebra H is AS-Gorenstein. Since a finite dimensional Hopf algebra H is Frobenius, injdimHH =injdimHH = 0, dim HomH(H❦, HH) = dim HomH(❦H, HH) = 1. Let g be a Lie algebra of dimension d. Then the universal enveloping algebra U(g) is AS-regular and of global dimension d. (Wu-Zhang) Noetherian affine PI Hopf algebras are AS-Gorenstein.

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Homological integrals of AS-Gorenstein algebras

(Lu, Wu and Zhang) Let A be an AS-Gorenstein algebra of injective dimension d. Then Extd

A(A❦, AA) is a 1-dimensional right A-module. Any

non-zero element in Extd

A(A❦, AA) is called a left homological

integral of A. We write l

A for Extd A(A❦, AA).

❦

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Homological integrals of AS-Gorenstein algebras

(Lu, Wu and Zhang) Let A be an AS-Gorenstein algebra of injective dimension d. Then Extd

A(A❦, AA) is a 1-dimensional right A-module. Any

non-zero element in Extd

A(A❦, AA) is called a left homological

integral of A. We write l

A for Extd A(A❦, AA).

Similarly, we have the right homological integrals. Extd

A(❦A, AA) is denoted by

r

A.

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Homological integrals of AS-Gorenstein algebras

(Lu, Wu and Zhang) Let A be an AS-Gorenstein algebra of injective dimension d. Then Extd

A(A❦, AA) is a 1-dimensional right A-module. Any

non-zero element in Extd

A(A❦, AA) is called a left homological

integral of A. We write l

A for Extd A(A❦, AA).

Similarly, we have the right homological integrals. Extd

A(❦A, AA) is denoted by

r

A.

l

A and

r

A are called left and right homological integral

modules of A respectively.

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Examples

If H is a finite dimensional Hopf algebra, then the homological integrals l and r are just the classical integrals of H. ❦

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Examples

If H is a finite dimensional Hopf algebra, then the homological integrals l and r are just the classical integrals of H. Let g be the 2-dimensional Lie algebra generated by x, y such that [x, y] = x. Let H = U(g). Then the right H-action on l is given by: for 0 = t ∈ l, t · x = 0 and t · y = −t. ❦

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Examples

If H is a finite dimensional Hopf algebra, then the homological integrals l and r are just the classical integrals of H. Let g be the 2-dimensional Lie algebra generated by x, y such that [x, y] = x. Let H = U(g). Then the right H-action on l is given by: for 0 = t ∈ l, t · x = 0 and t · y = −t. If g is a finite dimensional semisimple Lie algebra, then l

U(g) ∼

= ❦.

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Proposition 3 Let A be a Noetherian augmented algebra such that A is CY of dimension d. Then A is AS-regular and of global dimension d. In addition, l

A ∼

= ❦ as right A-modules.

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AS-Gorenstein Hopf algebras

Let A be a Hopf algebra, and ξ : A → ❦ an algebra

• homomorphism. We let [ξ] be the winding homomorphism of

ξ defined by [ξ](a) = ξ(a1)a2, for all a ∈ A. Proposition 4 (Brown-Zhang) Let A be a Noetherian AS-Gorenstein Hopf algebra with injective dimension d. Let l

A = ❦ξ, where ξ : A → ❦ is an algebra

• homomorphism. Then the rigid dualizing complex of A is [ξ]S2

AA[d]. 16 / 39

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AS-Gorenstein Hopf algebras

Theorem 5 (He-Van Oystaeyen-Zhang) Let A be a Noetherian AS-Gorenstein Hopf algebra. Then A is CY algebra of dimension d if and only if (ii) A is AS-regular with global dimension d and l

A ∼

= ❦ as right A-modules. (ii) S2

A is an inner automorphism of A.

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Pointed Hopf algebras

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Pointed Hopf algebras

A Hopf algebra A is called pointed, if all its simple left or right comodules are 1-dimensional. This is equivalent to saying that the coradical of A is a group algebra. ❦ ❦ ❦

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Pointed Hopf algebras

A Hopf algebra A is called pointed, if all its simple left or right comodules are 1-dimensional. This is equivalent to saying that the coradical of A is a group algebra. Examples: group algebras, universal enveloping algebras of Lie algebras, and quantized enveloping algebras of finite dimensional semisimple Lie algebras. ❦ ❦ ❦

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Pointed Hopf algebras

A Hopf algebra A is called pointed, if all its simple left or right comodules are 1-dimensional. This is equivalent to saying that the coradical of A is a group algebra. Examples: group algebras, universal enveloping algebras of Lie algebras, and quantized enveloping algebras of finite dimensional semisimple Lie algebras. For a pointed Hopf algebra A, its coradical filtration is a Hopf algebra filtration. ❦ ❦ ❦

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Pointed Hopf algebras

A Hopf algebra A is called pointed, if all its simple left or right comodules are 1-dimensional. This is equivalent to saying that the coradical of A is a group algebra. Examples: group algebras, universal enveloping algebras of Lie algebras, and quantized enveloping algebras of finite dimensional semisimple Lie algebras. For a pointed Hopf algebra A, its coradical filtration is a Hopf algebra filtration. Let Gr A be its associated graded Hopf algebra. Gr A ∼ = R#❦Γ, where ❦Γ is the coradical of A and R is a braided Hopf algebra in the category of Yetter-Drinfeld modules over ❦Γ.

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Nichols algebras

The vector space V consisting of primitive elements of R is a Yetter-Drinfeld module over ❦Γ.

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Nichols algebras

The vector space V consisting of primitive elements of R is a Yetter-Drinfeld module over ❦Γ. The algebra B(V ) generated by V is a braided Hopf subalgebra of R. It is called the Nichols algebra of V .

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Nichols algebras

The vector space V consisting of primitive elements of R is a Yetter-Drinfeld module over ❦Γ. The algebra B(V ) generated by V is a braided Hopf subalgebra of R. It is called the Nichols algebra of V . The algebra structure and coalgebra structure of B(V ) depend only on the braiding of V .

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Nichols algebras

The vector space V consisting of primitive elements of R is a Yetter-Drinfeld module over ❦Γ. The algebra B(V ) generated by V is a braided Hopf subalgebra of R. It is called the Nichols algebra of V . The algebra structure and coalgebra structure of B(V ) depend only on the braiding of V . Example: positive parts of U+

q (g) of the quantized enveloping

algebra of a finite dimensional semisimple Lie algebra g.

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Classification of pointed Hopf algebras

In general, the classification problem of pointed Hopf algebras has three parts: (i) Structure of the Nichols algebras B(V ). (ii) The lifting problem: Determine the structure of all pointed Hopf algebras A with G(A) = Γ such that Gr A ∼ = B(V )#❦Γ. (iii) Generation in degree one: Decide which Hopf algebras A are generated by group-like and skew-primitive elements.

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Pointed Hopf algebras U(D, λ)

The Hopf algebras U(D, λ) constructed by Andruskiewitsch and Schneider constitute a large class of pointed Hopf algebras with finite Gelfand-Kirillov dimension, whose group-like elements form an abelian group (arXive:math.QA/0110136).

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Pointed Hopf algebras U(D, λ)

The Hopf algebras U(D, λ) constructed by Andruskiewitsch and Schneider constitute a large class of pointed Hopf algebras with finite Gelfand-Kirillov dimension, whose group-like elements form an abelian group (arXive:math.QA/0110136). Such a pointed Hopf algebra U(D, λ) is viewed as a generalization of the quantized enveloping algebra Uq(g), where g is a finite dimensional semisimple Lie algebra.

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Γ: a free abelian group of finite rank s; ❦

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Γ: a free abelian group of finite rank s; D(Γ, (gi)1iθ, (χi)1iθ, (aij)1i,jθ): a generic datum of finite Cartan type for Γ. That is,

(aij) ∈ Zθ×θ is a Cartan matrix of finite type, where θ ∈ N; Let X be the set of connected components of the Dynkin diagram corresponding to the Cartan matrix (aij). If 1 i, j θ, then i ∼ j means that they belong to the same connected component; g1, · · · , gθ are elements in Γ and χ1, · · · , χθ are characters in Γ such that χj(gi)χi(gj) = χi(gi)aij, χi(gi) = 1 is not a root of unity, for all 1 i, j θ.

For simplicity, we define qij = χj(gi), 1 i, j θ. ❦

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Γ: a free abelian group of finite rank s; D(Γ, (gi)1iθ, (χi)1iθ, (aij)1i,jθ): a generic datum of finite Cartan type for Γ. That is,

(aij) ∈ Zθ×θ is a Cartan matrix of finite type, where θ ∈ N; Let X be the set of connected components of the Dynkin diagram corresponding to the Cartan matrix (aij). If 1 i, j θ, then i ∼ j means that they belong to the same connected component; g1, · · · , gθ are elements in Γ and χ1, · · · , χθ are characters in Γ such that χj(gi)χi(gj) = χi(gi)aij, χi(gi) = 1 is not a root of unity, for all 1 i, j θ.

For simplicity, we define qij = χj(gi), 1 i, j θ. λ: a family of linking parameters for D. That is, λ = (λij)1i<jθ is a family of elements in ❦ such that λij = 0 if i ≁ j, gigj = 1 and χiχj = ε.

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Given a datum D, we fix a braided vector space defined as follows. Let V be a Yetter-Drinfeld module over the group algebra ❦Γ with basis xi ∈ V χi

gi , 1 i θ. Then V is a braided vector

space of diagonal type whose braiding is given by c(xi ⊗ xj) = qijxj ⊗ xi, 1 i, j θ. The braiding is called generic if qii is not a root of unity for all 1 i θ. ❦ ❦ ❦

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Given a datum D, we fix a braided vector space defined as follows. Let V be a Yetter-Drinfeld module over the group algebra ❦Γ with basis xi ∈ V χi

gi , 1 i θ. Then V is a braided vector

space of diagonal type whose braiding is given by c(xi ⊗ xj) = qijxj ⊗ xi, 1 i, j θ. The braiding is called generic if qii is not a root of unity for all 1 i θ. The algebra U(D, λ) is defined to be the quotient Hopf algebra of the smash product ❦x1, · · · , xθ#❦Γ modulo the ideal generated by the following relations

(adc xi)1−aij(xj) = 0, 1 i, j θ, i = j, i ∼ j, xixj − χj(gi)xjxi = λij(1 − gigj), 1 i < j θ, i ≁ j,

where adc is the braided adjoint representation. ❦

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Given a datum D, we fix a braided vector space defined as follows. Let V be a Yetter-Drinfeld module over the group algebra ❦Γ with basis xi ∈ V χi

gi , 1 i θ. Then V is a braided vector

space of diagonal type whose braiding is given by c(xi ⊗ xj) = qijxj ⊗ xi, 1 i, j θ. The braiding is called generic if qii is not a root of unity for all 1 i θ. The algebra U(D, λ) is defined to be the quotient Hopf algebra of the smash product ❦x1, · · · , xθ#❦Γ modulo the ideal generated by the following relations

(adc xi)1−aij(xj) = 0, 1 i, j θ, i = j, i ∼ j, xixj − χj(gi)xjxi = λij(1 − gigj), 1 i < j θ, i ≁ j,

where adc is the braided adjoint representation. Gr U(D, λ) ∼ = U(D, 0) ∼ = B(V )#❦Γ.

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The Calabi-Yau property of U(D, λ)

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Homological integral of U(D, λ)

Theorem 6 (Yu-Zhang) Let D be a generic datum of finite Cartan type for a group Γ, λ a family of linking parameters for D, and A the Hopf algebra U(D, λ). Then A is Noetherian AS-regular and of global dimension p + s, where s is the rank of Γ and p is the number of the positive roots of the Cartan matrix. The left homological integral module l

A of A is isomorphic to ❦ξ,

where ξ : A → ❦ is an algebra homomorphism defined by ξ(g) = (p

i=1 χβi )(g) for all g ∈ Γ and ξ(xi) = 0 for all 1 i θ.

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Rigid Dualizing complex of U(D, λ)

Theorem 7 (Yu-Zhang) Let D be a generic datum of finite Cartan type for a group Γ, λ a family of linking parameters for D and A the Hopf algebra U(D, λ). (1) The rigid dualizing complex of the Hopf algebra A = U(D, λ) is ψA[p + s], where ψ is defined by ψ(xk) = p

i=1,i=jk χβi (gk)xk, for all 1 k θ, and

ψ(g) = (p

i=1 χβi )(g) for all g ∈ Γ where each jk, 1 k θ,

is the integer such that βjk = αk. (2) The algebra A is CY if and only if p

i=1 χβi = ε and S2 A is an

inner automorphism. Remark: For a pointed Hopf algebra U(D, λ), it is CY if and only if its associated graded algebra U(D, 0) is CY.

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Classification

In this classification, we assume that ❦ = C. CY pointed Hopf algebras U(D, λ) of dimension 3

Case Cartan matrix Generators Relations Case 1 trivial yh, y−1

h

y±1

h

y±1

m

= y±1

m

y±1

h

1 h 3 y±1

h

y∓1

h

= 1 1 h, m 3 Case 2 (I) A1 × A1 y±1

1

, x1, x2 y1y−1

1

= y−1

1

y1 = 1 y1x1 = qx1y1 y1x2 = q−1x2y1, 0 < |q| < 1 x1x2 − q−k x2x1 = 0, k ∈ Z+ Case 2 (II) A1 × A1 y±1

1

, x1, x2 y1y−1

1

= y−1

1

y1 = 1 y1x1 = qx1y1 y1x2 = q−1x2y1, 0 < |q| < 1 x1x2 − q−k x2x1 = (1 − y2k

1 ), k ∈ Z+

Remark: Uq(sl2) belongs to Case 2 (II). 28 / 39

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CY pointed Hopf algebras U(D, λ) of dimension 4

Case Cartan matrix Generators Relations Case 1 trivial yh, y−1

h

y±1

h

y±1

m

= y±1

m

y±1

h

1 h 4 y±1

h

y∓1

h

= 1 1 h, m 4 Case 2 (I) A1 × A1 y±1

1

, y±1

2

, x1, x2 y±1

h

y±1

m

= y±1

m

y±1

h

y±1

h

y∓1

h

= 1 1 h, m 2 y1x1 = q1 x1y1, y1x2 = q−1

1

x2y1 y2x1 = q2 x1y2, y2x2 = q−1

2

x2y2 0 < |q1 | < 1 x1x2 − q−k

1

x2x1 = 0, k ∈ Z+ Case 2 (II) A1 × A1 y±1

1

, y±1

2

, x1, x2 y±1

h

y±1

m

= y±1

m

y±1

h

y±1

h

y∓1

h

= 1 1 h, m 2 y1x1 = q1 x1y1, y1x2 = q−1

1

x2y1 y2x1 = q2 x1y2, y2x2 = q−1

2

x2y2 0 < |q1 | < 1 x1x2 − q−k

1

x2x1 = 1 − y2k

1 , k ∈ Z+

Let A and B be two algebras in Case (I) (or (II)) defined by triples (k, q1 , q2 ) and (k′, q′

1 , q′ 2 ) respectively. They

are isomorphic if and only if k = k′, q1 = q′

1 and there is some integer b, such that q′ 2 = qb 1 q2 or q′ 2 = qb 1 q−1 2

. 29 / 39

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Case 2 (III) A1 × A1 y±1

1

, y±1

2

, x1, x2 y±1

h

y±1

m

= y±1

m

y±1

h

y±1

h

y∓1

h

= 1 1 h, m 2 y1x1 = qx1y1, y1x2 = q−1x2y1 y2x1 = q

k l x1y2, y2x2 = q− k l x2y2

x1x2 − q−k x2x1 = 0 k, l ∈ Z+, 0 < |q| < 1 Case 2 (IV) A1 × A1 y±1

1

, y±1

2

, x1, x2 y±1

h

y±1

m

= y±1

m

y±1

h

y±1

h

y∓1

h

= 1 1 h, m 2 y1x1 = qx1y1, y1x2 = q−1x2y1 y2x1 = q

k l x1y2, y2x2 = q− k l x2y2

x1x2 − q−k x2x1 = 1 − yk

1 yl 2

k, l ∈ Z+, 0 < |q| < 1 30 / 39

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Case 2 (V) A1 × A1 y±1

1

, y±1

2

, x1, x2 y±1

h

y±1

m

= y±1

m

y±1

h

y±1

h

y∓1

h

= 1 1 h, m 2 y1x1 = qx1y1, y1x2 = q−1x2y1 y2x1 = q

k−l1 l2

x1y2, y2x2 = q

− k−l1 l2

x2y2 x1x2 − q−k x2x1 = 0 k, l1, l2 ∈ Z+, 0 < l1 < l2, 0 < |q| < 1 Case 2 (VI) A1 × A1 y±1

1

, y±1

2

, x1, x2 y±1

h

y±1

m

= y±1

m

y±1

h

y±1

h

y∓1

h

= 1 1 h, m 2 y1x1 = qx1y1, y1x2 = q−1x2y1 y2x1 = q

k−l1 l2

x1y2, y2x2 = q

− k−l1 l2

x2y2 x1x2 − q−k x2x1 = 1 − yk+l1

1

yl2

2

k, l1, l2 ∈ Z+, 0 < l1 < l2, 0 < |q| < 1 31 / 39

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Example

Let A be the algebra with generators xi, y±1

j

, 1 i, j 3, subject to the relations y±1

i

y±1

j

= y±1

j

y±1

i

, y±1

j

y∓1

j

= 1, 1 i, j 3, yj(xi) = χi(yj)xiyj, 1 i, j 3, x2

1x2 − qx1x2x1 − q2x1x2x1 + q3x2x2 1 = 0,

x2

2x1 − q−2x2x1x2 − q−1x2x1x2 + q−3x1x2 2 = 0,

x1x3 = x3x1. A is a CY pointed Hopf algebra of type A2 × A1 of dimension 7.

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SLIDE 59

Example

Let A be the algebra with generators xi, y±1

j

, 1 i, j 3, subject to the relations y±1

i

y±1

j

= y±1

j

y±1

i

, y±1

j

y∓1

j

= 1, 1 i, j 3, yj(xi) = χi(yj)xiyj, 1 i, j 3, x2

1x2 − qx1x2x1 − q2x1x2x1 + q3x2x2 1 = 0,

x2

2x1 − q−2x2x1x2 − q−1x2x1x2 + q−3x1x2 2 = 0,

x1x3 = x3x1. A is a CY pointed Hopf algebra of type A2 × A1 of dimension 7. The non-trivial liftings of A are also CY.

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SLIDE 60

The Calabi-Yau property of Nichols algebra of finite Cartan type

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SLIDE 61

Let D be a generic datum of finite Cartan type and λ a family

• f linking parameters for D.

❦ r ❦

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SLIDE 62

Let D be a generic datum of finite Cartan type and λ a family

• f linking parameters for D.

Gr U(D, λ) ∼ = U(D, 0) ∼ = B(V )#❦Γ. r ❦

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SLIDE 63

Let D be a generic datum of finite Cartan type and λ a family

• f linking parameters for D.

Gr U(D, λ) ∼ = U(D, 0) ∼ = B(V )#❦Γ. The Nichols algebra B(V ) is generated by xi, 1 i θ, subject to the relations adc(xi)1−aijxj = 0, 1 i, j θ, i = j. r ❦

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SLIDE 64

Let D be a generic datum of finite Cartan type and λ a family

• f linking parameters for D.

Gr U(D, λ) ∼ = U(D, 0) ∼ = B(V )#❦Γ. The Nichols algebra B(V ) is generated by xi, 1 i θ, subject to the relations adc(xi)1−aijxj = 0, 1 i, j θ, i = j. The Nichols algebra B(V ) is an Np+1-filtered algebra, whose associated graded algebra GrB(V ) is isomorphic to the following algebra: ❦xβ1, · · · , xβp | xβi xβj = χβj (gβi )xβj xβi , 1 i < j p, where xβ1, · · · , xβp are the root vectors of B(V ).

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SLIDE 65

The CY property of Nichols algebras

Theorem 8 (Yu-Zhang) Let V be a generic braided vector space of finite Cartan type, and R = B(V ) the Nichols algebra of V . For each 1 k θ, let jk be the integer such that βjk = αk. (1) The rigid dualizing complex is isomorphic to ϕR[p], where ϕ is the algebra automorphism defined by ϕ(xk) = (

jk−1

• i=1

χ−1

k (gβi ))( p

• i=jk+1

χβi (gk))xk =

p

• i=1,i=jk

χβi (gk)xk, for any 1 k θ.

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SLIDE 66

The CY property of Nichols algebras

(2) The algebra R is a CY algebra if and only if

jk−1

• i=1

χk(gβi ) =

p

• i=jk+1

χβi (gk), for any 1 k θ. Remark: The rigid dualizing complex of R = B(V ) is an Np+1-filtered bimodule, whose associated graded bimodule

GrϕGrR is isomorphic to the rigid dualizing complex of GrR.

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SLIDE 67

Gr U(D, λ) ∼ = U(D, 0) ∼ = B(V )#❦Γ.

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SLIDE 68

Gr U(D, λ) ∼ = U(D, 0) ∼ = B(V )#❦Γ. Proposition 9 If A = U(D, λ) is a CY algebra, then the rigid dualizing complex of the Nichols algebra R = B(V ) is isomorphic to ϕR[p], where ϕ is defined by ϕ(xk) = χ−1

k (gk)xk, for all 1 k θ.

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SLIDE 69

Gr U(D, λ) ∼ = U(D, 0) ∼ = B(V )#❦Γ. Proposition 9 If A = U(D, λ) is a CY algebra, then the rigid dualizing complex of the Nichols algebra R = B(V ) is isomorphic to ϕR[p], where ϕ is defined by ϕ(xk) = χ−1

k (gk)xk, for all 1 k θ.

Proposition 10 If the Nichols algebra R = B(V ) is a CY algebra, then the rigid dualizing complex of A = U(D, λ) is isomorphic to ψA[p + s], where ψ is defined by ψ(xk) = xk for all 1 k θ and ψ(g) = p

i=1 χβi (g) for all g ∈ Γ.

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SLIDE 70

Thank you!

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SLIDE 71

Thank you!

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